Feebly compact paratopological groups and real-valued functions

Sanchis, Manuel; Tkachenko, Mikhail

doi:10.1007/s00605-012-0444-3

Cited by 13 publications

(4 citation statements)

References 24 publications

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“…Proof. Let G be a feebly compact non-countably compact Baire Hausdorff Abelian paratopological group with all countable subsets closed constructed by Sanchis and Tkachenko in the proof of [35,Theorem 2]. They also constructed an open subsemigroup C of the group G such that C −1 is a closed discrete subspace of G and G = CC −1 .…”

Section: ]mentioning

confidence: 99%

Suitable sets for paratopological groups

Lin¹,

Ravsky²,

Shi³

2020

Preprint

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We investigate when a paratopological group G has a suitable set S. The latter means that S is a discrete subspace of G, S ∪ {e} is closed, and the subgroup S of G generated by S is dense in G.A paratopological group is a group G endowed with a topology τ such that the group operationIn this case τ is called a semigroup topology on G. If, additionally, the operation of taking inverse is continuous then G is a topological group. A classical example of a paratopological group failing to be a topological group is the Sorgenfrey line S, that is the additive group of real numbers, endowed with the Sorgenfrey topology generated by the base consisting of half-intervals [a, b), a < b).Whereas the investigation of topological groups already is a fundamental branch of topological algebra (see, for instance, [30], [12], and [2]), paratopological groups are not so well-studied and have more variable structure. Basic properties of paratopological groups compared with the properties of topological groups are described in the book [2] by Arhangel'skii and Tkachenko, in the PhD thesis of the second author [34], papers [32], [33], and in the survey [40] by Tkachenko.Suitable set for topological groups were considered by Hofmann and Morris in [23]. Fundamental results were obtained by Comfort et al.in [9] and Dikranjan et al. in [10] and in [11]. In the present paper we extend this research to paratopological groups.A subset S of a paratopological group G is a suitable set for G, if S is a discrete subspace of G, S ∪ {e} is closed, and the subgroup S of G generated by S is dense in G, see [20]. Let S (respectively, S c ) be the class of paratopological groups G having a suitable (respectively, closed suitable) set. It turns out that very often a suitable set of a group G generates G. This fact suggests to devote a special attention to classes S g (respectively, S cg ) of paratopological groups G having a suitable (respectively, closed suitable) set which generates G.Proposition 1. Any paratopological group with a suitable set is a T 1 -space or a twoelement group.Proof. Let S be a suitable set of a paratopological group G and s be any element of S. Put {s}. The definition of a suitable set implies {s} ⊂ {s, e}. If {s} or {e} is a

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Section: ]mentioning

confidence: 99%

Suitable sets for paratopological groups

Lin¹,

Ravsky²,

Shi³

2020

Preprint

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“…An opposite inclusion does not hold. Manuel Sanchis and Mikhail Tkachenko constructed a Hausdorff pseudocompact Baire not 2-pseudocompact paratopological group [SanTka,Th.2], answering author's questions from a previous version of the manuscript.…”

Section: Definitionsmentioning

confidence: 99%

Pseudocompact paratopological groups that are topological

Ravsky¹

2014

Preprint

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We obtain necessary and sufficient conditions when a pseudocompact paratopological group is topological. (2-)pseudocompact and countably compact paratopological groups that are not topological are constructed. It is proved that each 2-pseudocompact paratopological group is pseudocompact and that each Hausdorff σ-compact pseudocompact paratopological group is a compact topological group. Our particular attention is devoted to periodic and topologically periodic pseudocompact paratopological groups.

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“…Every compact space and every sequentially compact space are countably compact, every countably compact space is countably pracompact, and every countably pracompact space is pseudocompact (see [2]). We observe that pseudocompact spaces in topological literature also are called lightly compact or feebly compact (see [3,10,27]).…”

Section: Introductionmentioning

confidence: 99%

Pseudocompactness, Products, and Topological Brandt λ0 -Extensions of Semitopological Monoids

Гутік

Ravsky²

2017

J Math Sci

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In the paper we study the preservation of pseudocompactness (resp., countable compactness, sequential compactness, ω-boundedness, totally countable compactness, countable pracompactness, sequential pseudocompactness) by Tychonoff products of pseudocompact (and countably compact) topological Brandt λ 0 i -extensions of semitopological monoids with zero. In particular we show that ifa family of Hausdorff pseudocompact topological Brandt λ 0 i -extensions of pseudocompact semitopological monoids with zero such that the Tychonoff product {S i : i ∈ I } is a pseudocompact space then the direct product B 0 λi (S i ), τ 0 B(Si) : i ∈ I endowed with the Tychonoff topology is a Hausdorff pseudocompact semitopological semigroup. Date: September 24, 2018. 2010 Mathematics Subject Classification. Primary 22A15, 54H10. Key words and phrases. Semigroup, Brandt λ 0 -extension, semitopological semigroup, topological Brandt λ 0 -extension, pseudocompact space, countably compact space, countably pracompact space, sequentially compact space, ω-bounded space, totally countably compact space, countable pracompact space, sequential pseudocompact space.

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Feebly compact paratopological groups and real-valued functions

Cited by 13 publications

References 24 publications

Suitable sets for paratopological groups

Suitable sets for paratopological groups

Pseudocompact paratopological groups that are topological

Pseudocompactness, Products, and Topological Brandt λ0 -Extensions of Semitopological Monoids

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